Question 1: Calculate the value of 5^0. Explain why this is true based on the laws of exponents.

Answer: 1

Question 2: Prove that for any non-zero number a, a^0 = 1 using the laws of exponents.

Answer: Using the law a^m / a^n = a^{m-n}, set m=n to show a^n / a^n = 1, thus a^0 = 1.

Question 3: Evaluate 10^0 * 3^0 and justify your answer.

Answer: 1 * 1 = 1

Question 4: Construct an example to illustrate why 0^0 is undefined, providing mathematical reasoning.

Answer: Discuss the conflicting limits as x approaches zero in 0^x and x^0, showing that defining 0^0 as 1 leads to inconsistencies.

Question 5: Simplify the expression (2^3) * (2^0) using the laws of exponents.

Answer: 2^{3+0} = 2^3 = 8

Question 6: Is the statement 'Any number raised to the zero power equals zero' true or false? Justify your answer.

Answer: False

Question 7: Plot the graph of y = 2^x for x from -2 to 2 on the grid.

Answer: Students will draw an exponential curve passing through points (-2, 0.25), (0,1), (2,4).

Question 8: Construct a triangle on the grid where one vertex is at the origin and the other vertices satisfy the Zero Index rule for exponents.

Answer: Any triangle with points (0,0), (a,0), (0,b), where a and b are positive, with side lengths calculated accordingly.

Question 9: Given the expression 4^0, explain why the result is consistent with the laws of exponents, and identify common misconceptions.

Answer: Because 4^0 = 4^{n-n} = 1, and a common mistake is to think 0^0 is also 1, which is undefined.

Question 10: Identify and correct the mistake in the statement: 'Since 7^0 = 0, the zero index rule implies 7^0 = 0.'

Answer: The mistake is that 7^0 = 1, not 0. The zero index rule states a^0=1 for a≠0.

Question 11: Challenge: Derive the Zero Index rule using logarithms or limits, and discuss its validity.

Answer: Students will discuss the limit of a^x as x approaches 0 and show it approaches 1 for a > 0, justifying the rule.